# Equation of Straight Line in Plane/General Equation

## Theorem

A straight line $\mathcal L$ is the set of all $\tuple {x, y} \in \R^2$, where:

- $\alpha_1 x + \alpha_2 y = \beta$

where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.

## Proof

Let $y = \map f x$ be the equation of a straight line $\mathcal L$.

From Line in Plane is Straight iff Gradient is Constant, $\mathcal L$ has constant slope.

Thus the derivative of $y$ with respect to $x$ will be of the form:

- $y' = c$

Thus:

\(\displaystyle y\) | \(=\) | \(\displaystyle \int c \rd x\) | Fundamental Theorem of Calculus | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle c x + K\) | Primitive of Constant |

where $K$ is arbitrary.

Taking the equation:

- $\alpha_1 x + \alpha_2 y = \beta$

it can be seen that this can be expressed as:

- $y = - \dfrac {\alpha_1} {\alpha_2} x + \dfrac {\beta} {\alpha_2}$

thus demonstrating that $\alpha_1 x + \alpha_2 y = \beta$ is of the form $y = c x + K$ for some $c, K \in \R$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $10$: Formulas from Plane Analytic Geometry: $10.7$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates